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A008425
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Theta series of {D_6}* lattice.
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3
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1, 0, 12, 64, 60, 0, 160, 384, 252, 0, 312, 960, 544, 0, 960, 1664, 1020, 0, 876, 2880, 1560, 0, 2400, 4224, 2080, 0, 2040, 5248, 3264, 0, 4160, 7680, 4092, 0, 3480, 9984, 4380, 0, 7200, 10880, 6552, 0, 4608, 14784, 8160, 0, 10560, 17664, 8224, 0, 7812, 18560
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OFFSET
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0,3
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COMMENTS
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag New York, 1999, ISBN 978-1-4757-6568-7, p. 120.
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LINKS
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FORMULA
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Apparently, a(n) = Sum_{d|n} d^2*(4*(Kronecker(-4,n/d) - Kronecker(-4,d))), n > 0. - Ralf Stephan, Dec 31 2014
Expansion of phi(x^2)^6 + 64 * x^3 * psi(x^4)^6 in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Dec 14 2016
G.f. is a period 1 Fourier series that satisfies f(-1 / (8 t)) = 16 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A008428. - Michael Somos, Dec 14 2016
G.f.: theta_3(0, x^2)^6 + theta_2(0, x^2)^6.
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EXAMPLE
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G.f. = 1 + 12*x^2 + 64*x^3 + 60*x^4 + 160*x^6 + 384*x^7 + 252*x^8 + 312*x^10 + ...
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MATHEMATICA
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a[n_] := DivisorSum[n, #^2*(4*(KroneckerSymbol[-4, n/#]-KroneckerSymbol[-4, #]))&]; a[0]=1; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 06 2016, after Ralf Stephan *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2]^6 + EllipticTheta[ 2, 0, x^2]^6, {x, 0, n}]; (* Michael Somos, Dec 14 2016 *)
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PROG
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(PARI) {a(n) = if( n<1, n==0, 4 * sumdiv(n, d, d^2 * (kronecker(-4, n/d) - kronecker(-4, d))))}; /* Michael Somos, Dec 14 2016 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^5 / (eta(x^2 + A) * eta(x^8 + A))^2)^6 + 64 * x^3 * (eta(x^8 + A)^2/ eta(x^4 + A))^6, n))}; /* Michael Somos, Dec 14 2016 */
(PARI) {a(n) = my(G); if( n<0, 0, G = [2, 0, 0, 0, 0, 1; 0, 2, 0, 0, 0, 1; 0, 0, 2, 0, 0, 1; 0, 0, 0, 2, 0, 1; 0, 0, 0, 0, 2, 1; 1, 1, 1, 1, 1, 3]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, Dec 14 2016 */
(Magma) A := Basis( ModularForms( Gamma1(8), 3), 52); A[1] + 12*A[3] + 64*A[4] + 60*A[5] + 160*A[7]; /* Michael Somos, Dec 14 2016 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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